Problem 48 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the Mean Value Theorem to show that the graph of a concave up function \(f\) is always above its tangent line; that is, show that $$ f(x)>f(c)+f^{\prime}(c)(x-c), \quad x \neq c $$

Step-by-Step Solution

Answer

For concave up functions, \(f(x) > f(c) + f'(c)(x - c)\).

Step 1: Understanding the Mean Value Theorem

The Mean Value Theorem states that for a function \(f\) that is continuous on \([a, b]\) and differentiable on \((a, b)\), there exists at least one point \(c\in (a, b)\) such that \(f'(c)=\frac{f(b)-f(a)}{b-a}\). This theorem ensures there is at least one point where the instantaneous rate of change (slope of tangent) equals the average rate of change on the interval.

Step 2: Analyzing Concave Up Functions

A function \(f\) is concave up on an interval if its derivative \(f'\) is increasing on that interval or equivalently, if its second derivative \(f''\) is positive. This means that the slope of the tangent line (given by \(f'(x)\)) increases as \(x\) increases.

Step 3: Establishing the Tangent Line Equation

The equation of the tangent line to the function \(f\) at the point \(x = c\) is given by \(T(x) = f(c) + f'(c)(x - c))\). This equation represents a straight line with slope \(f'(c)\) passing through the point \((c, f(c))\).

Step 4: Comparing the Function and its Tangent Line

Considering that \(f\) is concave up, we have \(f''(x) > 0\) for \(x eq c\). This implies that beyond the point \(c\) in either direction, \(f(x)\) increases at a rate faster than the linear increase of \(T(x)\), since \(f'(x)\) itself is increasing. Thus, for \(x > c\) (or \(x < c\)), the function \(f(x)\) is getting further from the tangent line in the positive direction: \(f(x) > T(x) = f(c) + f'(c)(x-c)\).

Key Concepts

concave up function

In calculus, a function is described as "concave up" on an interval if it forms a distinctive upward curve on a graph. This occurs when the second derivative of a function, denoted as \(f''(x)\), is positive over the interval. A function that is concave up resembles the shape of an upwards-facing cup or bowl.
When analyzing a graph:

• The curve of a concave-up function always bends upwards.
• The rate of change, or slope, of the tangent lines to the curve continually increases.

This quality indicates that for a concave-up function, the original function \(f(x)\) grows at an accelerating rate as you move along the graph from left to right. It's a cornerstone concept in determining how a function graph behaves between given intervals.

tangent line

A tangent line represents a straight line that touches a function's curve at a single point, without crossing it at that location. In the context of calculus, the tangent line at a specific point \(c\) provides the instantaneous direction or slope of the curve. This slope hints at how the function is changing at that point.
To describe a tangent line accurately:

• The line is characterized by its slope, \(f'(c)\), which is the derivative of the function at point \(c\).
• The device of the tangent line equation takes the form \(T(x) = f(c) + f'(c)(x-c)\).

With this tangent line, you get a local linear approximation around \(c\), describing how rapidly the function value changes at that local point.

function derivative

A function derivative, denoted as \(f'(x)\), is a core principle of calculus reflecting the instantaneous rate of change of the function \(f(x)\). The derivative's value at a point tells you the slope of the tangent line to the curve at that particular point. Essentially, it describes how the output value of the function changes concerning changes in input.
Derivatives can:

• Indicate how a function increases or decreases at any specific point.
• Provide insights into the speed of change.

In the context of the Mean Value Theorem, the derivative assures that there's at least one point where the slope of the tangent line is equivalent to the average slope over a given interval \([a, b]\). Understanding derivatives well is crucial for exploring the overall behavior and characteristics of a function.

concavity analysis

Concavity analysis is essential in understanding the shape and nature of a function's graph. It tells us whether a function is curving upwards or downwards over a certain interval. This is determined by analyzing the sign of the second derivative \(f''(x)\).
For concave up:

• The second derivative \(f''(x)\) is greater than zero.
• This indicates that the slope of the tangent line is increasing, leading to the function being above its tangent lines, as it curves upwards.

Through concavity analysis, not only can you claim whether a function is concave up, but you can also predict how function values relate to the tangent lines, especially in demonstrating relations like having the function values consistently greater than their tangent approximations.